Prove that the given transformation is a linear transformation, using the definition. = Definition: A transformation T: R"→ R" is called a linear transformation if the following is true. 1. T(u + v) = T(u) + T(v) for all u and v in Rº 2. T(cv) = cT(v) for all v in R" and all scalars c -y x + 7y 8x - 5y Use the following remark. Remark: The definition of a linear transformation can be streamlined by combining (1) and (2) as shown below. T(C₁₂V₁ + C₂V₂) = C₁ T(v₁) + c₂ T(v₂) for all v₁, v₂ in R" and scalars C₁, C₂ T is a linear transformation if and only if T(C₁ V₁ + C₂ V₂) = C₂ T(v₁) + C₂ T(v₂), where v₁ = - [K] ([*]+[C]) T(C₂v₁ + C₂v₂) Then we get the following. 11 -(5₂81 +5₂Y₂) 41*1 (C₁x₂ + ₂x₂) + ( )(C₂x₂ + C₂Y₂) ]) ] )(c₁×₂ + €₂×₂) + (¯¯)(C₂Y/₂ + C₂Y/₂) -C1Y₁ ]) ₁₂x₁₂ + 1 +(C 14₁/₁ -Y₁ 1)9₁8/₁ V₂ +5₂₂ = [3]. -52/2 1) <₂*₂ + ( [ -1/₂ 1) ²₂/₂ ])/₂ X
Prove that the given transformation is a linear transformation, using the definition. = Definition: A transformation T: R"→ R" is called a linear transformation if the following is true. 1. T(u + v) = T(u) + T(v) for all u and v in Rº 2. T(cv) = cT(v) for all v in R" and all scalars c -y x + 7y 8x - 5y Use the following remark. Remark: The definition of a linear transformation can be streamlined by combining (1) and (2) as shown below. T(C₁₂V₁ + C₂V₂) = C₁ T(v₁) + c₂ T(v₂) for all v₁, v₂ in R" and scalars C₁, C₂ T is a linear transformation if and only if T(C₁ V₁ + C₂ V₂) = C₂ T(v₁) + C₂ T(v₂), where v₁ = - [K] ([*]+[C]) T(C₂v₁ + C₂v₂) Then we get the following. 11 -(5₂81 +5₂Y₂) 41*1 (C₁x₂ + ₂x₂) + ( )(C₂x₂ + C₂Y₂) ]) ] )(c₁×₂ + €₂×₂) + (¯¯)(C₂Y/₂ + C₂Y/₂) -C1Y₁ ]) ₁₂x₁₂ + 1 +(C 14₁/₁ -Y₁ 1)9₁8/₁ V₂ +5₂₂ = [3]. -52/2 1) <₂*₂ + ( [ -1/₂ 1) ²₂/₂ ])/₂ X
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Prove that the given transformation is a linear transformation, using the definition.
=
Definition: A transformation T: R"→ R" is called a linear transformation if the following is true.
1. T(u + v) = T(u) + T(v) for all u and v in Rº
2. T(cv) = cT(v) for all v in R" and all scalars c
-y
x + 7y
8x - 5y
Use the following remark.
Remark: The definition of a linear transformation can be streamlined by combining (1) and (2) as shown below.
T(C₁₂V₁ + C₂V₂) = C₁ T(v₁) + c₂ T(v₂) for all v₁, v₂ in R" and scalars C₁, C₂
T is a linear transformation if and only if
T(C₁ V₁ + C₂ V₂) = C₂ T(v₁) + C₂ T(v₂), where v₁ = - [K]
([*]+[C])
T(C₂v₁ + C₂v₂)
Then we get the following.
11
-(5₂81 +5₂Y₂)
41*1
(C₁x₂ + ₂x₂) + (
)(C₂x₂ + C₂Y₂)
])
] )(c₁×₂ + €₂×₂) + (¯¯)(C₂Y/₂ + C₂Y/₂)
-C1Y₁
]) ₁₂x₁₂ + 1
+(C
14₁/₁
-Y₁
1)9₁8/₁
V₂
+5₂₂
= [3].
-52/2
1) <₂*₂ + ( [
-1/₂
1) ²₂/₂
])/₂
X](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8fa7aa4-dafc-4c07-b274-462aa6ff800c%2Fdadebabc-c871-41df-be31-1819b5432991%2Foynl36p_processed.png&w=3840&q=75)
Transcribed Image Text:Prove that the given transformation is a linear transformation, using the definition.
=
Definition: A transformation T: R"→ R" is called a linear transformation if the following is true.
1. T(u + v) = T(u) + T(v) for all u and v in Rº
2. T(cv) = cT(v) for all v in R" and all scalars c
-y
x + 7y
8x - 5y
Use the following remark.
Remark: The definition of a linear transformation can be streamlined by combining (1) and (2) as shown below.
T(C₁₂V₁ + C₂V₂) = C₁ T(v₁) + c₂ T(v₂) for all v₁, v₂ in R" and scalars C₁, C₂
T is a linear transformation if and only if
T(C₁ V₁ + C₂ V₂) = C₂ T(v₁) + C₂ T(v₂), where v₁ = - [K]
([*]+[C])
T(C₂v₁ + C₂v₂)
Then we get the following.
11
-(5₂81 +5₂Y₂)
41*1
(C₁x₂ + ₂x₂) + (
)(C₂x₂ + C₂Y₂)
])
] )(c₁×₂ + €₂×₂) + (¯¯)(C₂Y/₂ + C₂Y/₂)
-C1Y₁
]) ₁₂x₁₂ + 1
+(C
14₁/₁
-Y₁
1)9₁8/₁
V₂
+5₂₂
= [3].
-52/2
1) <₂*₂ + ( [
-1/₂
1) ²₂/₂
])/₂
X
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